There are three main ways to earn Mode Points: listening to music, using Boosters, and getting Deals. Earn points from listening to music by going to the Music tab. Different activities will earn you your points in different ways - with Music you will earn per minute, doing Boosters will normally give you the points immediately after the activity you did (with a few exceptions). For Deals, it can take a few days or even a few weeks due to the verification processes that have to take place before you get your points. These offers are also subject to change/cancellation and are only available while inventory lasts.
Here's how to locate those tabs.